NON-SOLVABLE LIE GROUPS WITH NEGATIVE RICCI CURVATURE

Until a couple of years ago, the only known examples of Lie groups admitting left-invariant metrics with negative Ricci curvature were either solvable or semisimple.We use a general construction from a previous article of the second named author to produce a large number of examples with compact Levi factor. Given a compact semisimple real Lie algebra 𝔲 and a real representation π satisfying some technical properties, the construction returns a metric Lie algebra (𝔲, π) with negative Ricci operator. In this paper, when u is assumed to be simple, we prove that 𝔩(𝔲, π) admits a metric having negative Ricci curvature for all but finitely many finite-dimensional irreducible representations of 𝔲⨂ℝℂ, regarded as a real representation of 𝔲. We also prove in the last section a more general result where the nilradical is not abelian, as it is in every (𝔲, π).